Computational-chemistry formalisms: Born-Oppenheimer approximation, Hartree-Fock theory, density-functional theory, basis sets, and the treatment of electron correlation. Derives molecular observables from the Schrödinger equation.
quantum-chemistry
Born-Oppenheimer approximation
Approximation separating electronic and nuclear motion because nuclei (~1836× heavier than electrons) move on much longer timescales.…
Hartree-Fock theory
Mean-field approximation that represents the many-electron wavefunction as a single Slater determinant, enforcing Pauli antisymmetry but…
Density-functional theory (DFT)
Reformulation of electronic-structure theory using the electron density n(r) as primary variable (Hohenberg-Kohn theorems). Kohn-Sham…
Basis set
Finite set of analytic functions (typically Gaussian-type orbitals) used to expand molecular orbitals. Categories: minimal (STO-3G),…
Electron correlation
Difference between the exact non-relativistic electronic energy and the Hartree-Fock energy — the 'missing' correlation that arises from…
Kohn–Sham equations
Reformulation of DFT as a self-consistent single-particle problem: an auxiliary non-interacting system of electrons in an effective…
Coupled cluster CCSD(T)
Size-consistent correlated post-HF method with exponential Ansatz e^T|HF⟩. CCSD(T) 'gold standard' for single-reference chemistry; scales…
Multireference methods (CASSCF, MRCI, NEVPT2)
For near-degenerate states (bond breaking, excited states, TM complexes). CASSCF optimizes MOs + CI in active space; dynamic correlation…
DFT-D3/D4 dispersion corrections
Grimme's empirical pairwise correction E_disp=-Σ s_n C_n^AB f_damp(R^AB)/R^n. Restores long-range London forces absent from (semi)local…
Time-dependent DFT (TDDFT)
Runge-Gross theorem extends DFT to time-dependent external potentials; linear-response gives excitation energies. Cheap alternative to…
Born–Oppenheimer approximation (detail)
Separation of electronic/nuclear motion via m_e/M small; breakdown at conical intersections; nonadiabatic coupling vectors.
Conical intersections (CI)
Degeneracies of electronic states in 3N-8 branching space; drive ultrafast photophysics; Berry phase around loop.
Hartree–Fock SCF equations
F C = S C ε; orbital energies ε_i; Koopmans' theorem; DIIS convergence; Roothaan equations for closed shell.
MP2 (second-order Møller–Plesset)
E_MP2 = -Σ |⟨ij||ab⟩|²/(ε_a+ε_b-ε_i-ε_j); captures dynamical correlation; modest cost O(N⁵) with RI.
DFT Jacob's ladder
LDA → GGA → meta-GGA → hybrid → double-hybrid ascending accuracy (and cost); Perdew's ladder; B3LYP most-cited.
Gaussian basis sets (STO-3G → cc-pV5Z)
Pople (6-31G*) vs Dunning (cc-pVXZ) correlation-consistent; extrapolation to CBS limit; effective core potentials for heavy atoms.
QM/MM multiscale methods
Small QM active region embedded in MM environment; subtractive/additive coupling; 2013 Nobel (Karplus, Levitt, Warshel).
Continuum solvation (PCM, COSMO)
Solvent as dielectric continuum; cavity with surface charges; cheap vs explicit; errors for specific H-bonds; SMD reparameterization.
TD-DFT excited states
Casida equations; linear-response formalism; fails for charge-transfer, double excitations; range-separated hybrids improve.
EOM-CC excited states
Exp(T)|Ψ⟩ reference + R operator generates excitations; accurate for valence/Rydberg; EOM-EE, IP, EA flavors.
Fragment molecular orbital (FMO)
Divide large system into fragments; multi-layer embedding; scaling improvement for proteins, materials.
VQE & QPE for chemistry
Variational quantum eigensolver with parametrized ansatz on near-term hardware; QPE needs fault tolerance; chemistry benchmark for quantum…
Coupled-cluster (Cizek 1966)
J Cizek 1966 coupled-cluster ansatz e^T |HF>; modern CCSD(T) gold-standard 0.1 kcal/mol thermochem; Bartlett-Musial 2007 review.
DFT Kohn-Sham (1965)
Kohn-Sham 1965 (Nobel 1998) DFT KS-equations; modern B3LYP / PBE / SCAN; >40 years of computational-chemistry workhorse.
MP2 (Møller-Plesset 1934)
C Møller-M S Plesset 1934 perturbation-correction to HF; modern MP2/cc-pVTZ standard 2nd-order benchmark.
CASSCF (Roos 1980)
B Roos 1980 complete-active-space SCF; modern multireference NEVPT2 + CASPT2 for excited-states + transition-metals.
TDDFT (Casida 1995)
M Casida 1995 time-dependent DFT linear-response; modern excited-state CIS(D) + ADC(2); workhorse spectra <1 eV.
DFT-embedding (Wesolowski 1993)
T Wesolowski-A Warshel 1993 frozen-density-embedding; modern QM/MM + projector-based embedding-DFT subsystem-chemistry.